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Partial Differential Operators of Elliptic Type
Nirio Shimakura

Translations of Mathematical Monographs
1992; 288 pp; hardcover
Volume: 99
ISBN-10: 0-8218-4556-X
ISBN-13: 978-0-8218-4556-1
List Price: US$120
Member Price: US$96
Order Code: MMONO/99
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This book, which originally appeared in Japanese, was written for use in an undergraduate course or first year graduate course in partial differential equations and is likely to be of interest to researchers as well. This book presents a comprehensive study of the theory of elliptic partial differential operators. Beginning with the definitions of ellipticity for higher order operators, Shimakura discusses the Laplacian in Euclidean spaces, elementary solutions, smoothness of solutions, Vishik-Sobolev problems, the Schauder theory, and degenerate elliptic operators. The appendix covers such preliminaries as ordinary differential equations, Sobolev spaces, and maximum principles. Because elliptic operators arise in many areas, readers will appreciate this book for the way it brings together a variety of techniques that have arisen in different branches of mathematics.


First year graduate students specializing in partial differential equations, researchers in other fields of mathematics.


"The book was designed for seniors or first-year graduate students in Sendai specializing in elliptic partial differential equations, and will fill the same role admirably today in America or anywhere."

-- Mathematical Reviews

"Clear, well written, interesting; although it is addressed to (undergraduate or graduate) students, it is certainly useful for partial differential equations researchers as well."

-- Zentralblatt MATH

Table of Contents

  • Partial differential operators of elliptic type
  • The Laplacian in Euclidean spaces
  • Constructions and estimates of elementary solutions
  • Smoothness of solutions
  • Vishik-Sobolev problems
  • General boundary value problems
  • Schauder estimates and applications
  • Degenerate elliptic operators
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